EXTENDED GROUPS IN HAMILTONIAN MECHANICS: THE OSCILLATOR AND THE KEPLER PROBLEM. P. G. L. LEACH. (Received 24 September 1980).
/. Austral. Math. Soc. (Series B) 23 (1981), 173-186
APPLICATIONS OF THE LIE THEORY OF EXTENDED GROUPS IN HAMILTONIAN MECHANICS: THE OSCILLATOR AND THE KEPLER PROBLEM P. G. L. LEACH
(Received 24 September 1980) (Revised 22 December 1980)
Abstract
The method of the Lie theory of extended groups has recently been formulated for Hamiltonian mechanics in a manner which is consistent with the results obtained using the Newtonian equation of motion. Here the method is applied to the three-dimensional time-independent harmonic oscillator and to the classical Kepler problem. The expected constants of motion are obtained. Previously unobserved relations between generators and invariants are also noticed.
1. Introduction
There is, it would seem, a never-ending search for methods which provide a way of determining symmetries and invariants for dynamical systems. Two useful methods are Noether's theorem (in its various forms) and the method of the Lie theory of extended groups. The present application of both methods is to test systems for the presence of exact symmetries and, if they exist, to determine the associated constants of the motion. It would not greatly surprise us if in the future they were, in some sense, used to determine approximate symmetries, for example, of an adiabatic type. The two methods are based on the concept of an invariance under an infinitesimal transformation of the dynamical variables. For Noether's theorem, the object which is left invariant is the Action Integral and, for the Lie method, it is the equation(s) of motion. The latter method is less restrictive than the ©Copyright Australian Mathematical Society 1981 173
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P. G. L. Leach
[2]
former and provides a greater number of invariants and/or allows a more general class of problem to be treated. Before proceeding further it is proper that we define the type of infinitesimal transformation about which we speak. We are dealing with point transformations only. As an example of the greater generality of the Lie method we cite a one-dimensional linear system. Allowing point transformations only, Noether's theorem yields five generators of symmetry whereas the Lie method provides eight. However, the more serious failings of Noether's theorem occur when multi-dimensional systems are studied. It does not provide the Jauch-HillFradkin tensor for the harmonic oscillator nor the Runge-Lenz vector for the classical Kepler problem. The Lie method does [5], [6]. To repair this deficiency in Noether's theorem, the use of velocity-dependent transformations has been proposed [1], [4]. Certainly the constants mentioned above satisfy the equations obtained with the more general type of transformation. Unfortunately, the wider class of admissible transformations results in an infinite number of symmetries for which no systematic method of determination exists. The same fault applies to the Lie method if the inclusion of velocity-dependent transformations is allowed. However, the inclusion is not necessary for the Lie method since all the useful invariants may be found with coordinate-dependent transformations only. As far as we are aware, invariants of value in describing the motion are either linear or quadratic in the velocities (momenta). It is these invariants which we term useful here. The difficulties associated with the use of velocity-dependent transformations in Noether's theorem are delineated more fully in the Appendix. Until recently [7], the application of the Lie method in mechanics has been to the Newtonian equation of motion whereas Noether's theorem may be applied in either a Lagrangian or Hamiltonian context. Once the Newtonian results were obtained, the results could be translated into Hamiltonian form [2], but it seemed to us to be a messy approach. As many problems occur in a Hamiltonian framework, we judged it better to formulate the Lie method in the Hamiltonian framework in such a way that the results obtained would be consistent with the results for the corresponding Newtonian system when it exists. The formulation turned out to be very straightforward and we simply quote the results here. The operator y(q, p, 0 = £(q, 03/3/ + i,,(q, t)d/dq, + £,(* P- ')3/3/>,
(1.1)
is a generator of a one-parameter Lie group for a Hamiltonian H(q, p, /) with equations of motion qt - dH/dPi = 0
and
p, + dH/dq, = 0,
(1.2)
131
Lie theory in Hamiltonian mechanics
175
provided the action of the first extension of Y on equations (1.2) gives zero whenever equations (1.2) are satisfied. The first extension is
P. q, P, 0 = Y(q, p, t) + 7,P(q, p, /)3/9, = 0
(2.3)
#'> + Y(q, p, r)a(q, 0 = 0.
(2.4)
and On rearranging equation (2.3) we obtain Sl=!%V-ifkWkj/*i)pJ,
(2-5)
where [f'J] is the inverse offip that is, / % =«;.
(2.6)
This expression for J, is substituted into equation (2.4). The terms of third order in the momenta yield
(2 7)
Tih - °
-
and so £(q, /) = a(t) + bt{t)qt.
(2.8)
Collecting and rearranging the terms of second order in the momenta, f
x {8,A m + BJ&m) = 0, (2.9) where equation (2.6) has also been used. For equation (2.9), it is obvious that TJ is at most quadratic in q. From the particular case j{J = 6^ we get i?/(* 0 = H f t + cu{i)qj + d,(t).
(2.10)
There is no point, at this stage, in considering the coefficients of first and zero-th order powers of p since these contain the functions g,.(q, t). Had we allowed the/_,•(/) to be functions of q as well, equation (2.10) would have been much more complex. Such dependence would occur when curvilinear coordinates are used and may, as a separate topic, be worthy of discussion. However, the cartesian form which we are using here is sufficient for the present purpose. To round off this section we complement equations (2.8) and (2.10)
[s ]
Lie theory in Hamiltonian mechanics
177
with the expression for f,(q, p, t) for the case when^ y is S/y. It is ?,(q. P. ') = -4p, + bjqft + bjPjqt - bjPjPf + Cfjqj + ckjPj + dt.
(2.11)
3. Harmonic oscillator The three-dimensional Hamiltonian H =W
time-independent
harmonic
+ q2) = liPiPi +
